Simon Donaldson shocked the mathematical world during the 1980s with a series of papers on the structure of four-dimensional spaces. Researchers had produced a collection of results during the previous decade that outlined a general understanding of the properties of spaces of five or more dimensions, and of course, the cases of one- and two-dimensional spaces were well known. Ironically, three- and four-dimensional spaces were the hardest to interpret, even though they are the most applicable to physical space (if time is considered to be the fourth dimension). Great progress was made in three-dimensions by William Thurston,who received a Fields Medal in 1982 for his efforts. That same year, Donaldson published his most remarkable result: four-dimensional space has highly unusual properties that are found in no other dimension. Speaking on the occasion of Donaldson's presentation with the 1986 Fields Medal, Michael Atiyah commented, "When Donaldson produced his first few...